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primary decomposition theorem

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For example if we consider C[x,y], then the ideal (x^2,xy) can be written as (x^2,xy) = (x) \cap (x,y)^2 = (x) \cap (x^2,y) etc. These are distinct minimal primary decompositions.

I believe it is not true. See for example Steps in Commutative Algebra (Sharp), LMS texts, Cor. 4.18, or Ideals, Varieties, and Algorithms (Cox, Little, O'Shea). What is true is that the radicals of the ideals appearing are unique across all minimal primary decompositions.

Thank you. This is fixed.

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